Density is the amount of mass packed into each unit of volume, found by dividing mass by volume: D = mV. Its units combine the two, such as grams per cubic centimeter. A 40 g block with a volume of 16 cm³ has D = 40 ÷ 16 = 2.5 g/cm³. Population density works the same way: people per unit of area.
In Spanish
densidad
Also called
mass density, mass per unit volume, population density
Same volume, different mass: the cube with more mass packed into each cubic centimeter has the greater density. Cover a letter in the triangle to see how to find it.
Formal definition and notation
The density of an object or substance is the ratioD = mV of its mass m to its volumeV, for V > 0. For a uniform material it is the same for every sample, so it can identify a substance. More generally, a density is any quantity per unit of length, area or volume, such as persons per squaremile or BTUs per cubic foot.
Rearranged: m = D × V and V = mD. If the object is a shape, find its volume with the right geometry formula first.
Density examples, step by step
Example 1: Find the density
Warm-up
A block has a mass of 54 g and a volume of 20 cm³. What is its density?
D = mV = 5420.
54 ÷ 20 = 2.7.
Answer: The density is 2.7 g/cm³.
Example 2: Find the mass
Core
A metal has a density of 19.3 g/cm³. What is the mass of a 5 cm³ piece?
Solve the formula for mass: m = D × V.
m = 19.3 × 5 = 96.5.
Answer: The piece has a mass of 96.5 g.
Example 3: Density with a cylinder
Test-style
A cylindrical tank with radius 2 m and height 5 m is full of oil with a density of 900 kg/m³. What is the mass of the oil?
Volume of a cylinder: V = π r2h = π (2)2(5) = 20π m³.
Mass: m = D × V = 900 × 20π = 18,000π kg.
Approximate: 18,000π ≈ 56,549 kg.
Answer: The oil has a mass of 18,000π kg, about 56,549 kg.
Common mistakes
✗ MistakeDensity of a 30 g stone with volume 12 cm³: 1230 = 0.4 g/cm³.
✓ InsteadD = mV = 3012 = 2.5 g/cm³.
Mass goes on top. Check with units: the answer must be grams per cubic centimeter, so grams are in the numerator.
✗ MistakeA bigger piece of the same wood has a bigger density.
✓ InsteadA bigger piece has more mass and more volume; the ratio, the density, stays the same.
Density describes the material, not the size of the sample. Doubling the size doubles both parts of the fraction.
✗ MistakeA 2 kg object with volume 500 cm³: 2500 = 0.004 g/cm³.
✓ Instead2 kg is 2000 g, so 2000500 = 4 g/cm³.
Convert to matching units before dividing. Kilograms with cubic centimeters gives a number 1000 times too small for g/cm³.
Not to be confused with
Densityvs.Mass
Mass is how much matter an object has (grams); density is mass per unit of volume (grams per cm³). A big pillow has more mass than a coin but much less density.
volume is how much space an object takes up (cm³). Density divides the mass by that volume.
Quick check: 3 questions
Question 1 of 3
A liquid has a mass of 150 g and a volume of 120 mL. What is its density?
Show answer and explanation
Answer: C.150120 = 1.25 g/mL. 0.8 divides volume by mass, 270 adds, and 30 subtracts.
Question 2 of 3
A material has a density of 2.5 g/cm³. What is the volume of a 75 g piece?
Show answer and explanation
Answer: A.V = mD = 752.5 = 30 cm³. 187.5 multiplies, 0.033 divides the wrong way, and 77.5 adds.
Question 3 of 3
A town of 8,400 people covers a rectangle 4 miles by 3 miles. What is its population density?
Show answer and explanation
Answer: C. Area = 4 × 3 = 12 square miles, and 8400 ÷ 12 = 700. 2,100 divides by 4 only, 1,200 divides by the perimeter 7, and 600 divides by the perimeter 14.
Density: frequently asked questions
What is the formula for density?
Density equals mass divided by volume: D = mV. If you know two of the three quantities, you can find the third with m = D × V or V = mD. The units of density are a mass unit over a volume unit, such as grams per cubic centimeter or kilograms per cubic meter.
What is population density?
Population density is the number of people per unit of area, found by dividing the population by the area. A county with 45,000 people on 300 square miles has a population density of 150 people per square mile. It uses the same idea as mass density: an amount spread over a space.
Why does density matter in geometry?
Many modeling problems give a density and a shape. You find the volume or area with a geometry formula, then multiply by the density to get a total, such as the mass of water in a cylinder or the number of trees in a park. The high school geometry standards include density for exactly this reason.
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